5 research outputs found
A Stream Calculus of Bottomed Sequences for Real Number Computation
AbstractA calculus XPCF of 1⊥-sequences, which are infinite sequences of {0,1,⊥} with at most one copy of bottom, is proposed and investigated. It has applications in real number computation in that the unit interval I is topologically embedded in the set Σ⊥,1ω of 1⊥-sequences and a real function on I can be written as a program which inputs and outputs 1⊥-sequences. In XPCF, one defines a function on Σ⊥,1ω only by specifying its behaviors for the cases that the first digit is 0 and 1. Then, its value for a sequence starting with a bottom is calculated by taking the meet of the values for the sequences obtained by filling the bottom with 0 and 1. The validity of the reduction rule of this calculus is justified by the adequacy theorem to a domain-theoretic semantics. Some example programs including addition and multiplication are shown. Expressive powers of XPCF and related languages are also investigated
Measurable Cones and Stable, Measurable Functions
We define a notion of stable and measurable map between cones endowed with
measurability tests and show that it forms a cpo-enriched cartesian closed
category. This category gives a denotational model of an extension of PCF
supporting the main primitives of probabilistic functional programming, like
continuous and discrete probabilistic distributions, sampling, conditioning and
full recursion. We prove the soundness and adequacy of this model with respect
to a call-by-name operational semantics and give some examples of its
denotations